The part that was left out is the fact that conjugating is the group action of a group on itself. in his example, let's say there's some 3-cycle (123) the cube

The part that was left out is the fact that conjugating is the group action of a group on itself. in his example, let's say there's some 3-cycle (123) the cuber knows. But really, the cuber wants to transform like (479). Conjugation being equivalent to a group action implies the cuber only needs to find a group that permutes 1->4, 2->7, 3->9. Let's say one is found: (14)(27)(39)... gTg' => [(14)(27)(39)...](123)[(14)(27)(39)...]' = (479)